The invention generally relates to the field of computer programs and systems, and to the field of computer aided design (CAD), computer-aided engineering (CAE), modeling, and simulation. Specifically, the invention described herein relates to modeling/simulating the flow of fluids.
A specific use of embodiments of the invention is in the numerical simulation of hydraulic fracture. While it is not the only potential application for the invention, the review of background art presented herein will focus on hydraulic fracture—with emphasis on the modeling of fluid flow inside of fractures.
Hydraulic fracturing (commonly referred to as fracking) is the process of initiation and propagation of an underground crack by pumping fluid at relatively high flow rates and pressures. Hydraulic fracturing is desired for a variety of reasons, including enhanced oil and gas recovery deep below the earth. Field data from hydraulic fracturing operations exists primarily in the form of pressure response curves. It is difficult to define the actual hydraulic fracture geometry from this data alone. Therefore, analytical solutions and numerical simulations are used to evaluate and predict the location, direction, and extent of these hydraulic fractures.
The first simplified theoretical models for hydraulic fracturing were developed by Crittendon (1959) and Hubbert and Wills (1957). The methods of fracture mechanics were first applied to this problem by Barenblatt (1962). Pioneering works include papers by Perkins and Kern (1961) who adapted the classic elasticity plane strain crack solution to establish the so-called PK model and Geertsma and de Klerk (1969) who developed the KGD model. A significant amount of research has been carried out to obtain analytical solutions for different cases Detournay and Garagash (2003). Many of these early works are summarized in Valko and Economides (2001) and Adachi, et al. (2007).
However, because the analytical model and the empirical approaches cannot handle fractures of arbitrary shape and orientations, a fully 3D hydraulic fracture numerical simulator is vital to the petroleum industry. Boone and Ingraffea (1990) took a first step towards a fully coupled numerical solution general purpose hydraulic fracture code. This work was followed up with a commercial software product for modeling hydraulic fracture as discussed in Carter (2000).
Most references on numerical simulations of hydraulic fracture in the technical literature assume a form of Poiseuille flow between parallel plates or a cubic law (which is closely related to Poiseuille flow) for fluid equilibrium within an open fracture—either a natural fracture or a propagating fracture. Herein tangential flow within a fracture will be referred to as gap flow.
The validity of cubic fluid flow laws within geo-mechanics fractures was studied in depth in Witherspoon, et. al. (1980) and Tsang and Witherspoon (1981) and their results are widely referenced in the numerical hydraulic fracture literature. In a cubic flow law, the volumetric flow rate is proportional to the cube of the effective fracture aperture.
While studying various topics in numerical simulations of hydraulic fracture Lam and Cleary (1988), Sesetty and Ghassemi (2012), Chen, et al. (2009), Carrier and Granet (2012) all used simple cubic flow solutions for modeling gap flow in cohesive zone models. In this context simple means that the effective aperture was the mechanical fracture aperture.
Gordeliy and Peirce (2013) and Gordeliy and Peirce (2013) investigated the use of the eXtended Finite Element Method (XFEM) coupled with hydrodynamic equations for gap flow to investigate problems in hydraulic fracture. They also used a simple cubic flow solution for modeling gap flow.
In Zhang, et al. (2009) the authors investigated fluid driven fracture within networks of natural fractures. To model the gap flow they followed the approach of Tsang and Witherspoon (1981) and others and used a cubic flow equation in which the flow is proportional to the cube of the effective hydraulic aperture. In their work they postulated and solved an evolution law for the effective hydraulic aperture.
Similarly while applying the eXtended Finite Element Method (XFEM) to hydraulic fracture Mohammadnejad and Khoei (2013) and Mohammadnejad and Khoei (2013) also used cubic flow laws with solution varying permeability.
Provided herein below is a listing of referenced literature, the contents of which are herein incorporated by reference.    Crittendon, B. C., “The mechanics of design and interpretation of hydraulic fracture treatments”, Journal of Petroleum Technology, pp. 21-29, 1959.    Hubbert, M. K. and Wills, D. G., “Mechanics of hydraulic fracturing”, Journal of Petroleum Technology, vol. 9, no. 6, pp. 153-168, 1957.    Barenblatt, G. I., “The mathematical theory of equilibrium crack in brittle fracture”, Advances in Applied Mechanics, vol. 7, pp. 5-129, 1962.    Perkins, T. K. and Kern, L. R., “Widths of hydraulic fractures”, Journal of Petroleum Technology, vol. 13, no. 9, pp. 937-49, 1961.    Geertsma, J. and de Klerk, F., “A rapid method of predicting width and extent of hydraulically induced fractures”, Journal of Petroleum Technology, vol. 21, no. 12, pp. 1571-1581, 1969.    Detournay, E. and Garagash, D., “The tip region of a fluid-driven fracture in a permeable elastic solid”, Journal of Fluid Mechanics, vol. 494, pp. 1-32, 2003.    Valko, P., Economides, M. J., “Hydraulic Fracture Mechanics”, John Wiley & Sons, 2001.    Adachi, J., Siebrits, E., Peirce, and A., Desroches, J., “Computer simulation of hydraulic fractures”, International Journal of Rock Mechanics & Mining Sciences, Vol. 44, pp. 739-757, 2007.    Boone, T. J., and Ingraffea, R., “A numerical procedure for simulation of hydraulically-driven fracture propagation in poroelastic media”, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 14, pp. 27-47, 1990.    Carter, B. J, Desroches, J., Ingraffea, A. R. and Wawrzynek, P. A., “Simulating fully 3d hydraulic fracturing”, Modeling in Geomechanics, Ed. Zaman, Booker, and Gioda, Wiley Publishers, 730 p, 2000.    Witherspoon, P. A., Tsang, Y. W., Iwai, W. K., and, Gale, J. E., “Validity of cubic law for fluid flow in a deformable rock fracture”, Water Resources Research, vol. 16, no. 6, pp. 1016-1024, 1980.    Tsang, Y. W., and Witherspoon, P. A., “Hydromechanical behavior of a deformable rock fracture subject to normal stress”, Journal of Geophysical Research, vol. 86, pp. 9287-9298, 1981.    Lam, K. Y., and Cleary, M. P., “Three-dimensional fracture propagation under specified well-bore pressure”, Int. J. Numer. Anal. Meth. Geomech, Vol. 12, pp. 583-598, 1988.    Sesetty, V., and Ghassemi, A., “Modeling and analysis of stimulation for fracture network generation”, Proceedings, Thirty-Seventh Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, Calif., Jan. 30-Feb. 1, 2012.    Chen, Z., Bunger, A. P., Zhang, X., and Jeffrey, R. G., “Cohesive zone finite element-based modeling of hydraulic fractures”, Acta Mechanica Solida Sinica, vol. 22, no. 5, 2009.    Carrier, B., and Granet, S., “Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model”, Engineering Fracture Mechanics, vol. 79, 312-328, 2012.    Gordeliy, E., and Peirce, A., “Coupling schemes for modeling hydraulic fracture propagation using the XFEM”, Comput. Methods Appl. Mech. Engrg., vol. 253, pp. 305-322, 2013.    Gordeliy, E., and Peirce, A., “Implicit level set schemes for modeling hydraulic fracture using the XFEM”, Comput. Methods Appl. Mech. Engrg., vol. 266, pp. 125-143, 2013.    Zhang, X., Jeffrey, R. G., and Thiercelin, M., “Mechanics of fluid-driven fracture growth in naturally fractured reservoirs with simple network geometries”, Journal of Geophysical Research, Vol. 114, 2009.    Mohammadnejad, T., and Khoei, A. R., “Hydro-mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method”, Int. J. Numer. Anal. Meth. Geomech, Vol. 37, pp. 1247-1279, 2013.    Mohammadnej ad, T., and Khoei, A. R., “An extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model”, Finite Elements in Analysis and Design, vol. 73, pp. 77-95, 2013.
In their most general form, the Navier-Stokes equations (NSE) are a set of general three-dimension partial differential equations governing the motion of viscous fluids. A general form of these equations is:
      ρ    ⁢          Dv      Dt        =            -              ∇        p              +          µ      ⁢                          ⁢                        ∇          2                ⁢        v              +    f  Where ρ is the fluid density, v is the fluid velocity, Dv/Dt is the material derivative, ∇p is the pressure gradient, μ is the dynamic viscosity, and f is a body force.
In their general form, the NSE are complex, computationally expensive to solve, and typically require sophisticated software tools to generate accurate solutions. However, in common engineering applications simplified versions of NSE, such as Darcy flow, Poiseuille flow, and Stokes flow, which are easier and quicker to solve, are often used as approximations to the full NSE.
The prior approaches described hereinabove use a single type of cubic flow within the fractures. What varied during the simulations was the effective aperture. These existing simulation techniques thus rely upon one version of the simplified NSE, which is not ideal for simulating the flow of fluid in a variable environment. Thus, existing techniques can benefit from methods and systems to model the flow of fluid using more accurate and more computationally efficient methods.